5.1 The Nature of Sound Waves
119
string move transverse to its length. Even so, this wave can be expressed as a sum
of two equal sinusoidal waves traveling in opposite directions.
As can be seen from Eq. (5.8), the speed of sound in a material will be greater
for ‘stiffer’ materials and smaller if the mass density is greater. The first statement
comes about simply because a stiffer material has stronger inter-material forces,
allowing a disturbance to transmit its influence quicker than for a weaker force,
while the second statement follows from the inertial property of mass: It is harder
to get a more massive body to accelerate. (It is harder to change the motion of a
heavier football player!) As a sound wave passes through a material, the material
is sequentially displaced from equilibrium. Each local region can be observed to
oscillate with time. As each region is coupled to the next, energy is transported
through the material. For a sound wave, material only locally moves by wiggling,
while the wave energy can be carried a long distance through the material.
An initial oscillation at one frequency will force the same frequency of motion
into adjacent regions, although adjacent regions will not be synchronized in
displacement, since there must be a time lag in the transferred motion. This idea
works for all waves, including light. (Although the vitreous humor of your eyes
changes the wavelength of light, it does not change its color, since color is associated
with frequency.) As a wave passes from one material to another, or through a nonuniform material, the wavelength of the wave can change, since the wavelength will
depend on the wave speed.
A special solution to the wave equation is a sinusoidal wave,
ξ(x, t) = A sin ((2π(x/λ ∓ t/T ) + φ) ,
(5.18)
with a single ‘wavelength’ λ, where λ is defined to be the distance from the region
with one value of displacement to the next region with the same value, and a single
‘period’ T , where T is defined to be the time for the material at x to oscillate through
a complete cycle of motion.
A sinusoidal wave has a characteristic maximum displacement from equilibrium
called the ‘amplitude’ of the wave A. The argument of the sine function is an angle
called the ‘phase’ of the wave. The fixed angle φ is the initial phase (at x & t = 0).
The sign between the x term and the t term in the phase depends on whether the
wave is taken traveling toward increasing or decreasing x. A fixed displacement
point on the wave has constant phase, and moves according to x = ±(λ/T )t +
constant. Thus, v = λ/T is called the ‘phase velocity’ of the wave.
These sinusoidal solutions are also called ‘simple waves’, ‘monochromatic
waves’, or, for audible sounds, pure notes. Because of the connection of such sounds
with music, a single frequency wave is referred to as a ‘harmonic’, and the motion of
material with sinusoidal variation as harmonic oscillation. Sinusoidal solutions turn
out to be very useful to study because of the principle of linear superposition. We
will be able to add lots of them with different frequencies to form more complicated
waves.
Suppose we follow a point that has a fixed amplitude on a sinusoidal wave as
the wave carries the point forward. We will see that point move with the speed v
119
string move transverse to its length. Even so, this wave can be expressed as a sum
of two equal sinusoidal waves traveling in opposite directions.
As can be seen from Eq. (5.8), the speed of sound in a material will be greater
for ‘stiffer’ materials and smaller if the mass density is greater. The first statement
comes about simply because a stiffer material has stronger inter-material forces,
allowing a disturbance to transmit its influence quicker than for a weaker force,
while the second statement follows from the inertial property of mass: It is harder
to get a more massive body to accelerate. (It is harder to change the motion of a
heavier football player!) As a sound wave passes through a material, the material
is sequentially displaced from equilibrium. Each local region can be observed to
oscillate with time. As each region is coupled to the next, energy is transported
through the material. For a sound wave, material only locally moves by wiggling,
while the wave energy can be carried a long distance through the material.
An initial oscillation at one frequency will force the same frequency of motion
into adjacent regions, although adjacent regions will not be synchronized in
displacement, since there must be a time lag in the transferred motion. This idea
works for all waves, including light. (Although the vitreous humor of your eyes
changes the wavelength of light, it does not change its color, since color is associated
with frequency.) As a wave passes from one material to another, or through a nonuniform material, the wavelength of the wave can change, since the wavelength will
depend on the wave speed.
A special solution to the wave equation is a sinusoidal wave,
ξ(x, t) = A sin ((2π(x/λ ∓ t/T ) + φ) ,
(5.18)
with a single ‘wavelength’ λ, where λ is defined to be the distance from the region
with one value of displacement to the next region with the same value, and a single
‘period’ T , where T is defined to be the time for the material at x to oscillate through
a complete cycle of motion.
A sinusoidal wave has a characteristic maximum displacement from equilibrium
called the ‘amplitude’ of the wave A. The argument of the sine function is an angle
called the ‘phase’ of the wave. The fixed angle φ is the initial phase (at x & t = 0).
The sign between the x term and the t term in the phase depends on whether the
wave is taken traveling toward increasing or decreasing x. A fixed displacement
point on the wave has constant phase, and moves according to x = ±(λ/T )t +
constant. Thus, v = λ/T is called the ‘phase velocity’ of the wave.
These sinusoidal solutions are also called ‘simple waves’, ‘monochromatic
waves’, or, for audible sounds, pure notes. Because of the connection of such sounds
with music, a single frequency wave is referred to as a ‘harmonic’, and the motion of
material with sinusoidal variation as harmonic oscillation. Sinusoidal solutions turn
out to be very useful to study because of the principle of linear superposition. We
will be able to add lots of them with different frequencies to form more complicated
waves.
Suppose we follow a point that has a fixed amplitude on a sinusoidal wave as
the wave carries the point forward. We will see that point move with the speed v